Theorem sepnsepolem2 48256

Description: Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 48257. Proof could be shortened by 1 step using ssdisjdr 48194. (Contributed by Zhi Wang, 1-Sep-2024.)

Hypothesis

Ref	Expression
sepnsepolem2.1	⊢ (𝜑 → 𝐽 ∈ Top)

Assertion

Ref	Expression
sepnsepolem2	⊢ (𝜑 → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))

Distinct variable groups:   𝑦,𝐷   𝑦,𝐽   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐷(𝑥)   𝐽(𝑥)

Proof of Theorem sepnsepolem2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sepnsepolem2.1	. 2 ⊢ (𝜑 → 𝐽 ∈ Top)
2		id 22	. . 3 ⊢ (𝐽 ∈ Top → 𝐽 ∈ Top)
3		sslin 4236	. . . . 5 ⊢ (𝑧 ⊆ 𝑦 → (𝑥 ∩ 𝑧) ⊆ (𝑥 ∩ 𝑦))
4		sseq0 4404	. . . . . 6 ⊢ (((𝑥 ∩ 𝑧) ⊆ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∩ 𝑧) = ∅)
5	4	ex 411	. . . . 5 ⊢ ((𝑥 ∩ 𝑧) ⊆ (𝑥 ∩ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∩ 𝑧) = ∅))
6	3, 5	syl 17	. . . 4 ⊢ (𝑧 ⊆ 𝑦 → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∩ 𝑧) = ∅))
7	6	adantl 480	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ⊆ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∩ 𝑧) = ∅))
8		ineq2 4207	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝑧))
9	8	eqeq1d 2728	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ 𝑧) = ∅))
10	9	adantl 480	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑦 = 𝑧) → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ 𝑧) = ∅))
11	2, 7, 10	opnneieqv 48244	. 2 ⊢ (𝐽 ∈ Top → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
12	1, 11	syl 17	1 ⊢ (𝜑 → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ∃wrex 3060   ∩ cin 3946   ⊆ wss 3947  ∅c0 4325  ‘cfv 6554  Topctop 22886  neicnei 23092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-top 22887  df-nei 23093

This theorem is referenced by:  sepnsepo  48257